Russian Mathematical Surveys, 2025, Volume 80, Issue 5, Pages 911–913 DOI: https://doi.org/10.4213/rm10277e (Mi rm10277)

DOI: https://doi.org/10.4213/rm10277e

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The prominent mathematician Revaz Valer’yanovich Gamkrelidze, Doctor of Sciences, professor, academician of the Russian Academy of Sciences, Lenin Prize laureate, principal researcher in the Steklov Mathematical Institute of Russian Academy of Sciences, passed away on 5 May 2025.

He was born on 4 February 1927 in Kutaisi. In 1945 ge enrolled in the Faculty of Physics and Mathematics of Tbilisi University, but after his first year there transferred to the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University, from which he graduated in 1950. After completing his postgraduate studies and defending the Ph.D. thesis in 1953, where L. S. Pontryagin was his scientific advisor, till his last days Gamkrelidze worked in the Department of Differential Equations of the Steklov Mathematical Institute, and in the period from 1988 till 1997 he was the head of the department.

Gamkrelidze left a huge scientific heritage. In his first papers he obtained fundamental results on the topology of algebraic varieties by expressing the Chern classes of projective varieties in terms of projective invariants and by obtaining multidimensional generalizations of the classical Plücker formula. The crucial step in these results was representing the cycles dual to cohomological Chern classes as linear combinations of algebraic varieties [1].

Gamkrelidze was one of the founders of the modern mathematical theory of optimal control. The key result there, the maximum principle, was originally stated by Pontryagin as a conjecture. Gamkrelidze proved the principle for linear control systems and developed a complete theory of such systems [2]. In this, now classical, theory the main features of the subject were identified, namely the role of convexity, the importance of relay control, the nature of optimal synthesis, and the form of regularity conditions. The ‘condition of general position’ introduced by Gamkrelidze, and known in a simplified later version as the ‘rank condition for full controllability’, became extremely widespread both in mathematicas and applications. The linear theory developed by Gamkrelidze determined the appearance of the field for many years to come; its completeness and transparency still serve as models for researchers studying more complicated nonlinear systems.

Gamkrelidze obtained fundamental results for various generalizations of the classical problem of optimal control. He is the author of the first version of the maximum principle for problems with phase constraints [3]. In spite of the subsequent generalizations, this theorem is still very relevant because the regularity assumptions in it outline a class of problems for which the optimality conditions can be examined rather fully.

These results due to Gamkrelidze occupy a significant place in the famous monograph The mathematical theory of optimal processes [4] by four authors, L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, which was published in 1961 and became subsequently the foundation of the whole optimal control theory. In 1962 its authors were awarded the Lenin Prize for a cycle of papers on ordinary differential equations and their applications to optimal control and oscillation theory.

In the 1960s, after the discovery of the maximum principle, Gamkrelidze introduced and investigated the concept of optimal sliding regime, one of the central concepts in optimal control [5]–[7]. His theory included rigorous definitions of generalized controls and generalized trajectories. It did not merely extend the familiar concept of generalized curve, but also established an important approximation lemma stating that not only each individual generalized trajectory, but also each continuous family of trajectories of the convexified system can uniformly be approximated.

After completing, by the early 1970s, the development of the general theory of first-order conditions for optimality [8], Gamkrelidze switched to the study of singular extremals, which turned out to play a key role in the construction of optimal synthesis for nonlinear systems. Investigations of higher-order optimality conditions [9] led him to believe that an invariant coordinate-free approach was needed and the machinery of differential geometry had to be extended for applications to the new class of problems. This resulted in the development of the special ‘chronological calculus’, connecting dynamics with the commutation properties of the vector fields defining the control system [10]. The geometric methods in optimal control developed by Gamkrelidze with his students were recognized in our country and overseas and resulted in the creation of new lines of research in mathematical control theory and the theory of dynamical systems and in the formation of teams of researchers exploring them. An important step in this direction was creating the foundations of the feedback-invariant optimal control theory [11]. The history of the discovery of the Pontryagin maximum principle was thoroughly described in [12], one of Gamkrelidze’s last publications.

For decades Gamkrelidze conducted extensive publishing and editorial activities. For many years he was the editor-in-chief of the Matematika journal of abstracts, which was an important source of information for mathematicians in our country. He founded a number of mathematics journals, of which he then became the editor-in-chief. He launched the idea and was the publisher and a co-editor of the encyclopedic series Current Problems in Mathematics. Fundamental Directions. This magnificent series, translated by the Springer-Verlag publishing house under the title Encyclopaedia of Mathematical Sciences will always be remembered as a monument to the Moscow mathematical school during its heyday and will promote its ideas withinn the international mathematical community.

Gamkrelidze repeatedly read lecture courses for undergraduate and graduate students and young mathematicians. He fostered a number of talented researchers.

The bright memory of the prominent researcher and teacher will forever remain in our hearts.

Bibliography
1.	R. V. Gamkrelidze, “Chern cycles of complex algebraic manifolds”, Izv. Akad. Nauk SSSR Ser. Mat., 20:5 (1956), 685–706 (Russian)
2.	R. V. Gamkrelidze, “Theory of processes in linear systems which are optimal with respect to rapidity of action”, Izv. Akad. Nauk SSSR Ser. Mat., 22:4 (1958), 449–474 (Russian)
3.	R. V. Gamkrelidze, “Optimal control processes for bounded phase coordinates”, Izv. Akad. Nauk SSSR Ser. Mat., 24:3 (1960), 315–356 (Russian)
4.	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The mathematical theory of optimal processes, Intersci. Publ. John Wiley & Sons, Inc., New York–London, 1962, viii+360 pp.
5.	R. V. Gamkrelidze, “Optimal sliding states”, Soviet Math. Dokl., 3 (1962), 559–562
6.	R. V. Gamkrelidze, “On some extremal problems in the theory of differential equations with applications to the theory of optimal control”, J. SIAM Control Ser. A, 3 (1965), 106–128
7.	R. V. Gamkrelidze, Principles of optimal control theory, Math. Concepts Methods Sci. Eng., 7, Plenum Press, New York–London, 1978, xii+175 pp.
8.	R. V. Gamkrelidze, “Necessary first order conditions, and the axiomatics of extremal problems”, Proc. Steklov Inst. Math., 112 (1971), 156–186
9.	A. A. Agračev (Agrachev) and R. V. Gamkrelidze, “A second order optimality principle for a time-optimal problem”, Math. USSR-Sb., 29:4 (1976), 547–576
10.	A. A. Agračev (Agrachev) and R. V. Gamkrelidze, “The exponential representation of flows and the chronological calculus”, Math. USSR-Sb., 35:6 (1979), 727–785
11.	A. A. Agrachev and R. V. Gamkrelidze, “Feedback-invariant optimal control theory and differential geometry. I. Regular extremals”, J. Dynam. Control Systems, 3:3 (1997), 343–389
12.	R. V. Gamkrelidze, “History of the discovery of the Pontryagin maximum principle”, Proc. Steklov Inst. Math., 304 (2019), 1–7

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\pages 911--913
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